,X such that V ∩ Y = C j so they are equal as a homology class of P 2 k + 1 Σ ,X ,i.e., [ V ∩ Y ] = [ C j ] . It is easy to check that π ( V ) ∩ X = C j as a subvariety of P k + 2 Σ where π ∶ ( x, y ) ↦ x . Hence [ π ( V ) ∩ X ] = [ C j ] which is equivalent to say that λ C j comes from P k + 2 Σ which contradicts the choice of [ C j ] .\n\nRemark 5.2 . Into the proof of the previous theorem, the key fact was that on X the Hodge conjecture holds and we translate it to Y by contradiction. So, using an analogous argument we have:\n\nargument we have:\n\nProposition 5.3. Let Y = { F = y 1 f s +⋯+ y s f s = 0 } ⊂ P 2 k + 1 Σ